Hey who said it can't be both? Character wakes up with kinda small udders, the go to work because that's not that big a deal and work is important.
Throughout the day they progressively get bigger and bigger until their clothes start to soak and the teats are too erect to properly keep the udders covered,,

They refuse to milk themself because that would be indecent! So they elect to go about their day as 'normal'. Everybody at their work is trying to hide their obvious erections and wetness, until they give in and just start milking the character.
The character is super embarrassed but can't do anything because their udders are too big and they're immobile...

The coworkers are loving it but the character is super red and hiding their face while still unable to stop the euphoric release of all the pent up milk~